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An optimization problem with volume constraint for an inhomogeneous operator with nonstandard growth
Schrödinger equations with vanishing potentials involving Brezis-Kamin type problems
1. | Departamento de Matemática, Universidade Federal de Sergipe, São Cristóvão-SE, 49100-000, Brazil |
2. | Departamento de Matematica y C. C., Universidad de Santiago de Chile, Casilla 307, Correo 2, Santiago, Chile |
3. | Unidade Acadêmica de Matemática, Universidade Federal de Campina Grande, Campina Grande 58429-900, Brazil |
$ \begin{equation*} -\Delta u+V(x)u = f(x,u),\,\,\, x\in\mathbb{R}^n,\,\, n\geq 3, \end{equation*} $ |
$ V $ |
$ f $ |
$ f(x,u) $ |
$ f $ |
$ \rho(x)f(u) $ |
$ f $ |
$ \rho $ |
$ \mathrm{(H)} $ |
$ \rho $ |
References:
[1] |
A. Ambrosetti, H. Brezis and G. Cerami,
Combined effects of concave and convex nonlinearities in some elliptic problems, J. Funct. Anal., 122 (1994), 519-543.
doi: 10.1006/jfan.1994.1078. |
[2] |
A. Ambrosetti, V. Felli and A. Malchiodi,
Ground states of nonlinear Schrödinger equations with potentials vanishing at infinity, J. Eur. Math. Soc., 7 (2005), 117-144.
doi: 10.4171/JEMS/24. |
[3] |
A. Bahrouni, H. Ounaies and V. D. Rădulescu,
Bound state solutions of sublinear Schrödinger equations with lack of compactness, RACSAM, 113 (2019), 1191-1210.
doi: 10.1007/s13398-018-0541-9. |
[4] |
A. Bahrouni, H. Ounaies and V. D. Rădulescu,
Infinitely many solutions for a class of sublinear Schrödinger equations with indefinite potentials, Proc. Roy. Soc. Edinburgh Sect. A, 145 (2015), 445-465.
doi: 10.1017/S0308210513001169. |
[5] |
H. Brezis and L. Oswald,
Remarks on sublinear elliptic equations, Nonlinear Analysis. Theory, Methods & Applications., 1 (1986), 55-64.
doi: 10.1016/0362-546X(86)90011-8. |
[6] |
H. Brezis and S. Kamin,
Sublinear elliptic equations in $\mathbb{R}^N$, Manuscripta Math., 74 (1992), 87-106.
doi: 10.1007/BF02567660. |
[7] |
H. Brezis and L. Nirenberg,
Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents, Comm. Pure Appl. Math., 36 (1983), 437-477.
doi: 10.1002/cpa.3160360405. |
[8] |
J. Chabrowski and J. M. B. do Ó,
On semilinear elliptic equations involving concave and convex nonlinearities, Math. Nachr., 233/234 (2002), 55-76.
doi: 10.1002/1522-2616(200201)233:1<55::AID-MANA55>3.0.CO;2-R. |
[9] |
D. G. de Figueiredo, J-P Gossez and P. Ubilla,
Local superlinearity and sublinearity for indefinite semilinear elliptic problems, J. Funct. Anal., 199 (2003), 452-467.
doi: 10.1016/S0022-1236(02)00060-5. |
[10] |
D. G. de Figueiredo, J-P Gossez and P. Ubilla,
Multiplicity results for a family of semilinear elliptic problems under local superlinearity and sublinearity, J. Eur. Math. Soc., 8 (2006), 269-286.
doi: 10.4171/JEMS/52. |
[11] |
F. Gazzola and A. Malchiodi,
Some remark on the equation $-\Delta u = \lambda(1+u)^p$ for varying $\lambda, p$ and varying domains, Comm. Partial Differential Equations, 27 (2002), 809-845.
doi: 10.1081/PDE-120002875. |
[12] |
D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, 1983.
doi: 10.1007/978-3-642-61798-0. |
[13] |
Q. Han and F. Lin, Elliptic Partial Differential Equations, Courant Lect. Notes Math., vol. 1, AMS, Providence, RI, 1997. |
[14] |
T-S Hsu and H-L Lin,
Four positive solutions of semilinear elliptic equations involving concave and convex nonlinearities in $\mathbb{R}^n$, J. Math. Anal. Appl., 365 (2010), 758-775.
doi: 10.1016/j.jmaa.2009.12.004. |
[15] |
Z. Liu and Z-Q Wang,
Schrödinger equations with concave and convex nonlinearities, Z. angew. Math. Phys., 56 (2005), 609-629.
doi: 10.1007/s00033-005-3115-6. |
[16] |
M. H. Protter and H. F. Weinberger, Maximum Principle in Differential Equations, Prentice Hall, Englewoood Cliffs, New Jersey, 1967. |
[17] |
T-F Wu,
Multiple positive solutions for a class of concave-convex elliptic problems in $\mathbb{R}^n$ involving sign-changing weight, J. Funct. Anal., 258 (2010), 99-131.
doi: 10.1016/j.jfa.2009.08.005. |
show all references
References:
[1] |
A. Ambrosetti, H. Brezis and G. Cerami,
Combined effects of concave and convex nonlinearities in some elliptic problems, J. Funct. Anal., 122 (1994), 519-543.
doi: 10.1006/jfan.1994.1078. |
[2] |
A. Ambrosetti, V. Felli and A. Malchiodi,
Ground states of nonlinear Schrödinger equations with potentials vanishing at infinity, J. Eur. Math. Soc., 7 (2005), 117-144.
doi: 10.4171/JEMS/24. |
[3] |
A. Bahrouni, H. Ounaies and V. D. Rădulescu,
Bound state solutions of sublinear Schrödinger equations with lack of compactness, RACSAM, 113 (2019), 1191-1210.
doi: 10.1007/s13398-018-0541-9. |
[4] |
A. Bahrouni, H. Ounaies and V. D. Rădulescu,
Infinitely many solutions for a class of sublinear Schrödinger equations with indefinite potentials, Proc. Roy. Soc. Edinburgh Sect. A, 145 (2015), 445-465.
doi: 10.1017/S0308210513001169. |
[5] |
H. Brezis and L. Oswald,
Remarks on sublinear elliptic equations, Nonlinear Analysis. Theory, Methods & Applications., 1 (1986), 55-64.
doi: 10.1016/0362-546X(86)90011-8. |
[6] |
H. Brezis and S. Kamin,
Sublinear elliptic equations in $\mathbb{R}^N$, Manuscripta Math., 74 (1992), 87-106.
doi: 10.1007/BF02567660. |
[7] |
H. Brezis and L. Nirenberg,
Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents, Comm. Pure Appl. Math., 36 (1983), 437-477.
doi: 10.1002/cpa.3160360405. |
[8] |
J. Chabrowski and J. M. B. do Ó,
On semilinear elliptic equations involving concave and convex nonlinearities, Math. Nachr., 233/234 (2002), 55-76.
doi: 10.1002/1522-2616(200201)233:1<55::AID-MANA55>3.0.CO;2-R. |
[9] |
D. G. de Figueiredo, J-P Gossez and P. Ubilla,
Local superlinearity and sublinearity for indefinite semilinear elliptic problems, J. Funct. Anal., 199 (2003), 452-467.
doi: 10.1016/S0022-1236(02)00060-5. |
[10] |
D. G. de Figueiredo, J-P Gossez and P. Ubilla,
Multiplicity results for a family of semilinear elliptic problems under local superlinearity and sublinearity, J. Eur. Math. Soc., 8 (2006), 269-286.
doi: 10.4171/JEMS/52. |
[11] |
F. Gazzola and A. Malchiodi,
Some remark on the equation $-\Delta u = \lambda(1+u)^p$ for varying $\lambda, p$ and varying domains, Comm. Partial Differential Equations, 27 (2002), 809-845.
doi: 10.1081/PDE-120002875. |
[12] |
D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, 1983.
doi: 10.1007/978-3-642-61798-0. |
[13] |
Q. Han and F. Lin, Elliptic Partial Differential Equations, Courant Lect. Notes Math., vol. 1, AMS, Providence, RI, 1997. |
[14] |
T-S Hsu and H-L Lin,
Four positive solutions of semilinear elliptic equations involving concave and convex nonlinearities in $\mathbb{R}^n$, J. Math. Anal. Appl., 365 (2010), 758-775.
doi: 10.1016/j.jmaa.2009.12.004. |
[15] |
Z. Liu and Z-Q Wang,
Schrödinger equations with concave and convex nonlinearities, Z. angew. Math. Phys., 56 (2005), 609-629.
doi: 10.1007/s00033-005-3115-6. |
[16] |
M. H. Protter and H. F. Weinberger, Maximum Principle in Differential Equations, Prentice Hall, Englewoood Cliffs, New Jersey, 1967. |
[17] |
T-F Wu,
Multiple positive solutions for a class of concave-convex elliptic problems in $\mathbb{R}^n$ involving sign-changing weight, J. Funct. Anal., 258 (2010), 99-131.
doi: 10.1016/j.jfa.2009.08.005. |
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